It fascinates me that Hillel’s seems to have won the most favor and respect (even from Jesus, no less!), not because he was on the “correct” side while Shammai was “wrong” about everything; but instead Hillel gets the commendation of having better embodied wisdom because he apparently did the better job of not dismissing what his opponent taught. I.e. - he was better able to step outside of himself and into his opponent’s shoes to “steel man” his opponent’s positions. (My own choice of words there.) Apparently Shammai lacked or shunned that particular practice of trying to inhabit his opponent’s point of view. For my own part - I benefit from engaging in the practice of trying to “channel” the priorities and thoughts of people who I view as political adversaries. I have a friend or two with whom I can practice this - and they too can practice channeling my own sides concerns back to me and we can give feedback to each other on how we did. And we’re both wiser for it I think - but more importantly, it’s helped to seal our relationship as trusted friends with a confidence that our relationship is above and immune to the differences we can rehearse to each other.
It may not be directly answering your query about how to ‘teach from contradiction’, but here is one excerpt I’ll quote directly from the end of that chapter that I thought interesting regarding the whole Shammai vs. Hillel tussle.
Early followers of Jesus benefited from the more tolerant spirit of the Hillel rabbinic school. When members of the Sanhedrin at Jerusalem wanted to kill Peter and John, Rabbi Gamaliel–grandson of Hillel–spoke up and said, “Keep away from these men and let them alone; because if this plan or this undertaking is of human origin, it will fail; but if it is of God, you wil not be able to overthrow them–in that case you may even be found fighting against God!” (Acts 5:38-39).
Later, Paul tells the Sanhedrin that he was brought up “at the feet of Gamaliel, educated strictly according to our ancestral law” (Acts 22:3). Paul’s early ardor for persecuting Christians suggests that he indeed was “educated strictly,” taught by Gamaliel but was drawn to the comparatively rigid theology of Shammai. After experiencing the grace of God in Jesus Christ, the apostle seems–like Jesus himself–to resonate with the spirit of Hillel.